Is it possible to vectorise a CNN?
Yes you can vectorize a CNN. See this github file for details: https://github.com/parasdahal/deepnet/blob/master/deepnet/layers.py

After looking through it it basically transposes the input to some dimension and apply matrix multiplication to the weight with some other kind of transfromation. Pls refer to the github repository for details. 

Hope this can help you and have a nice day!